This invariant point is a necessity to deploy a linear representation of the \(O(4)\)  group around it, just as in Eq. (REF ), and hence to write down HEFT as a SMEFT.
It happens that the existence of such zero of \(\mathcal {F}(h)\)  is not a sufficient condition for SMEFT to be possible. It may happen that non-analyticities arise at the fixed point \(h_\ast \)  or between that \(h_\ast \)  and \(h=0\)  (the physical vacuum), spoiling the possibility of constructing a viable SMEFT around \(h_\ast \)  that is applicable in particle physics. This is why we must require that the metric and thus \(\mathcal {F}(h)\)  be analytic in a sufficient domain (see [1]} for further detail).
In a lower energy regime \(m_h\sim E\)  where the potential \(V\)  makes a relevant contribution, the same considerations also apply to \(V\) .
